For the radius ratio $\eta\,{\equiv}\,R_i/R_o\,{=}\,0.1$ and several rotation rate ratios $\mu\,{\equiv}\,\Omega_o/\Omega_i$, we consider the linear stability of spiral Poiseuille flow (SPF) up to ${\hbox {\it Re}}\,{=}\,10^5$, where $R_i$ and $R_o$ are the radii of the inner and outer cylinders, respectively, ${\hbox {\it Re}}\,{\equiv}\,\overline V_Z(R_o\,{-}R_i)/\nu$ is the Reynolds number, $\Omega_i$ and $\Omega_o$ are the (signed) angular speeds of the inner and outer cylinders, respectively, $\nu$ is the kinematic viscosity, and $\overline V_Z$ is the mean axial velocity. The Re range extends more than three orders of magnitude beyond that considered in the previous $\mu\,{=}\,0$ work of Recktenwald et al. (Phys. Rev. E, vol. 48, 1993, p. 444). We show that in the non-rotating limit of annular Poiseuille flow, linear instability does not occur below a critical radius ratio $\hat\eta\,{\approx}\,0.115$. We also establish the connection of the linear stability of annular Poiseuille flow for $0\,{<}\,\eta\,{\leq}\,\hat\eta$ at all Re to the linear stability of circular Poiseuille flow ($\eta\,{=}\,0$) at all Re. For the rotating case, with $\mu\,{=}\,{-}1$, ${-}\,0.5$, ${-}\,0.25$, 0 and 0.2, the stability boundaries, presented in terms of critical Taylor number ${\hbox {\it Ta}}\,{\equiv}\,\Omega_i(R_o\,{-}R_i)^2/\nu$ versus Re, show that the results are qualitatively different from those at larger $\eta$. For each $\mu$, the centrifugal instability at small Re does not connect to a high-Re Tollmien–Schlichting-like instability of annular Poiseuille flow, since the latter instability does not exist for $\eta\,{<}\,\hat\eta$. We find a range of Re for which disconnected neutral curves exist in the $k$–Ta plane, which for each non-zero $\mu$ considered, lead to a multi-valued stability boundary, corresponding to two disjoint ranges of stable Ta. For each counter-rotating ($\mu\,{<}\,0$) case, there is a finite range of Re for which there exist three critical values of Ta, with the upper branch emanating from the ${\hbox {\it Re}}\,{=}\,0$ instability of Couette flow. For the co-rotating ($\mu\,{=}\,0.2$) case, there are two critical values of Ta for each Re in an apparently semi-infinite range of Re, with neither branch of the stability boundary intersecting the Re = 0 axis, consistent with the classical result of Synge that Couette flow is stable with respect to all small disturbances if $\mu\,{>}\,\eta^2$, and our earlier results for $\mu\,{>}\,\eta^2$ at larger $\eta$.

The properties of adjoint operators and the method of composite expansion are used to study the generation of Tollmien–Schlichting (TS) waves in the leading-edge region of an incompressible, flat-plate boundary layer. Following the classical asymptotic approach, the flow field is divided into an initial receptivity region, where the unsteady motion is governed by the linearized unsteady boundary-layer equation (LUBLE), and a downstream linear amplification area, where the evolution of the unstable mode is described by the classical Orr–Sommerfeld equation (OSE). The large $\bar{x}$ behaviour of the LUBLE is analysed using a multiple-scale expansion which leads to a set of composite differential equations uniformly valid in the wall-normal direction. These are solved numerically as an eigenvalue problem to determine the local properties of the Lam and Rott eigensolutions. The receptivity coefficient for an impinging acoustic wave is extracted by projecting the numerical solution of the LUBLE onto the adjoint of the Lam and Rott eigenfunction which, further downstream, turns into an unstable TS wave. In the linear amplification region, the main characteristics of the instability are recovered by using a multiple-scale expansion of the Navier–Stokes equations and solving numerically the derived eigenvalue problems. A new matching procedure, based on the properties of the adjoint Orr–Sommerfeld operator, is then used to check the existence and the extent of an overlapping domain between the two asymptotic regions. Results for different frequencies are discussed.

We investigate spatial and temporal cross-correlations between streamwise and normal velocity components in three shear flows: a low-dimensional model for vortex–streak interactions, direct numerical simulations for a nearly homogeneous shear flow and experimental data for a turbulent boundary layer. Driving of streamwise streaks by streamwise vortices gives rise to a temporal asymmetry in the short-time correlation. Close to the wall or the bounding surface in the free-slip situations, this asymmetry is identified. Further away from the boundaries the asymmetry becomes weaker and changes character, indicating the prevalence of other processes. The systematic variation of the asymmetry measure may be used as a complementary indicator to separate different layers in turbulent shear flows. The location of the extrema at different streamwise displacements can be used to read off the mean advection speed; it differs from the mean streamwise velocity because of asymmetries in the normal extent of the structures.

The subgrid-scale (SGS) stress in the atmospheric surface layer is studied using measurement data. Field measurements employing a novel array technique were conducted to provide data for obtaining resolvable- and subgrid-scale variables. We analyse the conditional SGS stress and the conditional stress production rate conditional on the resolvable-scale velocity, which must be reproduced by the SGS model for large-eddy simulation (LES) to predict correctly the one-point resolvable-scale velocity statistics. The results show that both buoyancy and shear play important roles in the physics of the SGS stress. Strong buoyancy and vertical shear associated with updrafts and positive streamwise velocity fluctuations cause conditional forward energy transfer and strong anisotropy in the conditional SGS stress. Downward returning flows associated with large convective eddies result in conditional energy backscatter and much less anisotropic SGS stress. Predictions of the conditional SGS stress and the conditional stress production rate predicted using several SGS models are compared with measurements. None of those models are able to predict correctly the trends of both statistics. The Smagorinsky and one nonlinear model under-predict the anisotropy and the variations of the anisotropy, whereas the other nonlinear model and the mixed model over-predict both. The deficiencies of the SGS models that cause inaccurate LES statistics, such as the over-prediction of the mean shear and under-prediction of the vertical velocity skewness, are identified. The present study shows that analyses of conditional SGS stress and conditional SGS stress production provide a systematic approach for studying SGS physics and evaluating SGS models and can potentially be used to target specific aspects of LES that are important for a given application.

Darwin (1953) introduced a simple heuristic that relates the Lagrangian fluid drift induced by a solid body propagating in irrotational flow to its virtual- or added-mass. The force required to accelerate the solid body must also overcome this added-mass. An extension of Darwin's (1953) method to the case of vortices propagating in a real fluid is described here. Experiments are conducted to demonstrate the existence of an added-mass effect during uni-directional vortex motion, which is analogous to the effect of solid bodies in potential flow. The definition of the vortex added-mass coefficient is modified from the solid body case to account for entrainment of ambient fluid by the vortex. This modified coefficient for propagating vortices is shown to be equal in magnitude to the classical coefficient for a solid body of equivalent boundary geometry. An implication of these results is that the vortex added-mass concept can be used as a surrogate for the velocity potential, in order to facilitate calculations of the pressure contribution to forces required to set fluid into unsteady vortical motion. Application of these results to unsteady wake analyses and fluid–structure interactions such as vortex-induced vibrations is suggested.

Observational data in the atmosphere indicate that conventionally defined drag and heat transfer coefficients increase rapidly as wind speed falls. It is shown here that, at sufficiently low wind speeds, the observed heat flux is nearly independent of wind speed but the drag increases linearly with it. These findings are not consistent with the free-convection limit of the Businger relations for Monin–Obukhov theory, and lend support to the ideas of Ingersoll (1966) and Grachev (1990), till now checked only against laboratory experiments. We propose here that it is useful to define, within the regime of mixed convection, a sub-regime of ‘weakly forced convection’ in which, to a first approximation, the heat flux is determined by temperature differentials as in free convection and the momentum flux by a perturbation, linear in wind, on free convection. It is further proposed that this regime is governed by velocity scales determined by the heat flux (rather than by the friction velocity as in classical turbulent boundary layer theory). Three candidates for the heat-flux velocity scale are considered; novel definitions of the drag and heat exchange coefficients, based on the preferred scale, are found to show very weak dependence on wind speed up to values of about 5–10 m s$^{-1}$; but there is some evidence that the usefulness of heat-flux scaling may extend beyond the velocity limits where pure free-convection scaling for heat flux is valid.

The description of entrainment in turbulent free jets is at the heart of physical models of some major flows in environmental science, from volcanic plumes to the dispersal of pollutant wastes. The classical approach relies on the assumption of complete self-similarity in the flows, which allows a simple parameterization of the dynamical variables in terms of constant scaling factors, but this hypothesis remains under debate. We use in this paper an original parameterization of entrainment and an extensive review of published experimental data to interpret the discrepancy between laboratory results in terms of the systematic evolution of the dynamic similarity of the flow as a function of downstream distance from the source. We show that both jets and plumes show a variety of local states of partial self-similarity in accordance with the theoretical analysis of George (1989), but that their global evolution tends to complete self-similarity via a universal route. Plumes reach this asymptotic regime faster than jets which suggests that buoyancy plays a role in more efficiently exciting large-scale modes of turbulence.

This work is an experimental study of the motion and deformation of a bioartificial capsule flowing in a tube of 4 mm diameter. The capsules, initially designed for medical applications, are droplets of salt water surrounded by a thin polymeric membrane. They are immersed in a very viscous Newtonian silicone oil that flows through a tube in the Stokes regime. The properties of the capsules were carefully determined. Two previous experimental papers were devoted to their characterization by osmotic swelling and compression between two plates. The present work also provides a series of tests that allows an accurate definition of the experimental model under investigation. The capsules are buoyant and initially quasi-spherical. Nevertheless, buoyancy and small departures from sphericity are shown to have no significant effects, provided the flowing velocity is large enough for the viscous stress to become predominant. The capsules are also initially slightly over-inflated, but there is no mass transfer through the membrane during the present experiments. Their volume therefore remains constant. The membrane can be described as an elastic two-dimensional material, the elastic moduli of which are independent of the deformation. Far from the tube ends, the capsule reaches a steady state that depends on two parameters: the capillary number, $\hbox{\it Ca}$; and the ratio of the radius of the capsule to that of the tube, $a/R$. The capillary number, which compares the hydrodynamic stresses to the elastic tensions in the membrane, was varied between 0 and 0.125. The radius ratio, which measures the magnitude of the confinement, was varied from 0.75 to 0.95. In the range investigated, the membrane material always remains in the elastic domain. At fixed $a/R$, the capsule is stretched in the axial direction when $\hbox{\it Ca}$ is increased. The process of deformation involves two main stages. At small to moderate $\hbox{\it Ca}$, the lateral dimension of the capsule decreases whereas its axial length increases. The capsule is rounded at both ends, but the curvature of its rear decreases as $\hbox{\it Ca}$ increases. At large $\hbox{\it Ca}$, the rear buckles inward. Then, the negative rear curvature goes on decreasing whereas the lateral dimension of the capsule reaches a constant value. On the other hand, increasing $a/R$ promotes the deformation: the process remains qualitatively the same, but the different stages are attained for smaller values of $\hbox{\it Ca}$. Comparisons with available numerical simulations show that the results are strongly dependent on the properties of the capsules.

The linear stability of two-dimensional surface gravity waves on fluid of finite depth is investigated for superharmonic disturbances. For this problem, Zufiria & Saffman (Stud. Appl. Maths vol. 74, 1986, p. 259) suggested that an exchange of stability occurs when the total wave energy becomes stationary as a function of wave speed for fixed ‘Bernoulli constant’. In defining the potential energy of the above total wave energy, the surface displacement was measured from the quiescent surface with the same ‘Bernoulli constant’. We have re-examined this problem both analytically and numerically, and found that the above ‘Bernoulli constant’ must be replaced by ‘mean surface height’ for the statement to be valid.

We analyse the flows in fluid-conveying tubes whose elastic walls perform small-amplitude high-frequency oscillations. We show that the velocity perturbations induced by the wall motion are dominated by their transverse components and use numerical simulations to analyse the two-dimensional flows that develop in the tube's cross-sections. Asymptotic methods are then employed to derive explicit predictions for the flow fields and for the total viscous dissipation, whose magnitude plays an important role in the development of self-excited oscillations.

We show that in cases with fluid–structure interaction, the coupled oscillations are controlled by the ratio of the fluid and wall densities, and by a material parameter that is equivalent to the Womersley number, and indicates the importance of fluid inertia and wall elasticity relative to the fluid's viscosity. We present numerical simulations of the coupled oscillations and use asymptotic techniques to derive explicit predictions for their period and decay rate. Finally, we discuss the implications of our results for the development of self-excited oscillations in three-dimensional collapsible tubes.

A drop translating in the presence of a uniform electric field is studied both theoretically and experimentally to determine qualitative properties of three-dimensional chaotic particle trajectories and mixing in bounded Stokes flows. The flow is a combination of a Hadamard–Rybczynski and a Taylor circulation due to the translation and electric field respectively. The three-dimensional trajectories are generated by tilting the electric field relative to the drop translational motion by an angle $\alpha$. The numerical analysis includes qualitative analysis of the degree of mixing by Poincaré mapping, quantitative estimates of the largest mixed volume fraction and the rate of mixing characterized by the largest Lyapunov exponent. Experiments are performed using a castor oil/silicone oil system for the continuous and dispersed phases respectively. Single trajectories are studied by visualizing small neutrally buoyant glass particles inside the dispersed phase using a stereoscopic particle tracking technique. Drops are approximately 5 mm in diameter, settling velocities are $O$(0.1 mm s$^{-1}$) and the electric fields are $O$(10 V mm$^{-1}$). We observe crossings of the unperturbed separatrix and particle trajectories that show evidence of a symmetry plane, both important features of the theory.

The self-similar solution that characterizes the water impact, with a constant vertical velocity, of a wedge entering the free surface with an arbitrary orientation is derived analytically. The study is carried out by assuming the fluid to be ideal, weightless and with negligible surface tension effects. The solution is based on the complex analysis of nonlinear two-dimensional problems of unsteady free boundary flows and is written in terms of two governing functions, which are the complex velocity and the derivative of the complex potential defined in a parameter domain. The boundary value problem is reduced to the system of an integral and an integro-differential equation in terms of the velocity modulus and of the velocity angle to the free surface, both written as functions of a parameter variable. The system of equations is solved through a numerical procedure which is validated in the case of symmetric wedges. Comparisons with data available in literature are established for this purpose. Results are presented in terms of free surface shape, contact angles at the intersection with the wedge boundary, pressure distribution, force and moment coefficients. For a given wedge angle, the changes induced by the heel angle on the above quantities are discussed. A criterion is proposed to determine the limit conditions beyond which flow separation from the wedge apex occurs. Comparisons with experimental results available in literature are presented.

This paper addresses the slender laminar flow resulting from the discharge of a low-Mach-number hot gas jet of radius $a$ and moderately large Reynolds number $R_j$ into a cold atmosphere of the same gas. We give the boundary-layer solution for plane and round jets with very small values of the ambient-to-jet temperature ratio $\varepsilon$ accounting for the temperature dependence of the viscosity and conductivity typical of real gases. It is seen that the leading-order description of the jet in the limit $\varepsilon \rightarrow 0$ exhibits a front-like structure, including a precisely defined separating boundary at which heat conduction and viscous shear stresses vanish in the first approximation, so that the temperature and axial velocity remain unperturbed outside the jet. Separate analyses are given for the jet discharging into a stagnant atmosphere, when the jet boundary is a conductive front, and for the jet discharging into a coflowing stream, when the jet boundary appears as a contact surface. We provide in particular the numerical description of the jet development region corresponding to axial distances of order $R_j a$ for buoyant and non-buoyant jets, as well as the self-similar solutions that emerge both in the near field and in the far field. In all cases considered, comparisons with numerical integrations of the boundary-layer problem for moderately small values of $\varepsilon$ indicate that these front descriptions give excellent predictions for the temperature and velocity fields in the near-axis region.

The integral momentum and tracer equations for the mean motion with the turbulence contribution in momentum and tracer fluxes are integrated on the centreline of either plane or round buoyant jets, using suitable assumptions for the spreading coefficients and a closing function, and unified first- and second-order solutions are derived in the entire buoyancy range for mean axial velocities and mean concentrations. Comparisons to experimental data in the literature validate the model and show that second-order solutions deviate less than first-order solutions. Both types are used in conjunction with the integral continuity and kinetic energy equations for the mean motion to determine the variation of the local Richardson and Froude numbers, dispersion ratio, bulk dilution, dilution ratio, entrainment coefficient and mean velocity, kinetic energy flux and its gradient for the mean motion; and the variations of these quantities are evaluated using reported experimental or theoretical data. Finally, the variation of the product of kinetic energy flux and the local Richardson number is examined and a universal constant for both plane and round buoyant jets is revealed, leading to a unified definition of the local Richardson number, which is independent of the flow and mixing geometry and could be useful. Simple computational programming and good overall agreement make the proposed model a very promising tool for laboratory and field studies, outfall design and validation of numerical models.

The motion of vortex sheets is susceptible to the onset of the Kelvin–Helmholz instability. There is now a large body of evidence that the instability leads to the formation of a curvature singularity in finite time. Vortex blob methods provide a regularization for the motion of vortex sheets. Instead of forming a curvature singularity in finite time, the curves generated by vortex blob methods form spirals. Theory states that these spirals will converge to a classical weak solution of the Euler equations as the blob size vanishes. This theory assumes that the blob method is the result of a convolution of the sheet velocity with an appropriate choice of a smoothing function. We consider four different blob methods, two resulting from appropriate choices of smoothing functions and two not. Numerical results indicate that the curves generated by these methods form different spirals, but all approach the same weak limit as the blob size vanishes. By scaling distances and time appropriately with blob size, the family of spirals generated by different blob sizes collapse almost perfectly to a single spiral. This observation is the next step in developing an asymptotic theory to describe the nature of the weak solution in detail.

Models for understanding and controlling oscillations in the flow past a rectangular cavity are developed. These models may be used to guide control designs, to understand performance limits of feedback, and to interpret experimental results. Traditionally, cavity oscillations are assumed to be self-sustained: no external disturbances are necessary to maintain the oscillations, and amplitudes are limited by nonlinearities. We present experimental data which suggests that in some regimes, the oscillations may not be self-sustained, but lightly damped: oscillations are sustained by external forcing, such as boundary-layer turbulence. In these regimes, linear models suffice to describe the behaviour, and the final amplitude of oscillations depends on the characteristics of the external disturbances. These linear models are particularly appropriate for describing cavities in which feedback has been used for noise suppression, as the oscillations are small and nonlinearities are less likely to be important. It is shown that increasing the gain too much in such feedback control experiments can lead to a peak-splitting phenomenon, which is explained by the linear models. Fundamental performance limits indicate that peak splitting is likely to occur for narrow-bandwidth actuators and controllers.

The effects of an oscillatory zero-net-mass-flux jet (i.e. synthetic jet) acting on a separated flow over a hump are investigated in terms of two actuation parameters – actuator position and forcing frequency. By considering the vorticity flux balance and introducing a centroid of vorticity production over the hump surface, lift and drag acting on the hump can be expressed as a function of this centroid and the rate of vorticity production. To study the parametric dependence of lift and drag, direct numerical simulation (DNS) is performed by solving compressible, unsteady, laminar flows over a half-cylindrical hump in two dimensions. The DNS results show that periodic actuation significantly reduces the rate of vorticity production at the wall and shifts the centroid upstream so that the drag is reduced and the lift is increased, respectively. When the actuation parameters are varied, it is found that the lift is governed by the horizontal coordinate of the vorticity-production centroid, while the drag is determined by both the vertical coordinate of the centroid and the rate of vorticity production over the hump. This paper explains by using ideal flow models that the vorticity-production centroid is controlled by two factors: one is the actuator position at which clockwise vorticity is generated, and the other is the point where the separation vortex is pinched off, thereby the clockwise vorticity being absorbed.

Simulations in a three-dimensional numerical wave tank are performed to investigate the shoaling and breaking of solitary waves over a sloping ridge. The numerical model solves fully nonlinear potential flow equations with a high-order boundary-element method combined with an explicit time-integration method, expressed in a mixed Eulerian–Lagrangian formulation. Analyses of shoaling and breaking-wave profiles and kinematics (both on the free surface and within the flow) are carried out. It is observed that the transverse modulation of the ridge topography induces three-dimensional effects on the time evolution, shape and kinematics of breaking waves. Comparisons of two- and three-dimensional results in the middle cross-section of the ridge, however, show remarkable similarities, especially for the shape and dynamics of the plunging jet.

Rotating grid turbulence experiments have been carried out in a stably stratified fluid for relatively large Reynolds numbers (mesh Reynolds numbers up to 18000). Under the combined effects of rotation and stratification the flow degenerates into quasi-horizontal motions. This regime is investigated using a scanning imaging velocimetry technique which provides time-resolved velocity fields in a volume. The most obvious effect of rotation is the inhibition of the kinetic energy decay, in agreement with the quasi-geostrophic model which predicts the absence of a direct energy cascade, as found in two-dimensional turbulence. In the regime of small Froude and Rossby numbers, the dynamics is found to be non-dissipative and associated with a symmetric and highly intermittent vertical vorticity field, that displays $k_h^{-3}$ energy spectra. For higher Rossby numbers, fundamental differences with the quasi-geostrophic model are found. A significant decay of kinetic energy, which does not depend on the stratification, is observed. Moreover, in this regime, although both cyclones and anticyclones are initially produced, the intense vortices are only cyclones. For late times the flow consists of an assembly of coherent interacting structures. Under the influence of both rotation and stratification, they take the form of lens-like eddies with aspect ratio proportional to $f/N$.

Direct and large-eddy simulations of the interaction between a laminar boundary layer and a von Kármán vortex street behind a circular cylinder are carried out for three values of the Reynolds number based on the cylinder diameter and free-stream velocity: $\hbox{\it Re}_D\,{=}\,385$, 1155 and 3900. Rapid, bypass-like transition to turbulence is observed in the two higher-Reynolds-number cases. Flow statistics in the transitional and turbulent regions are examined, followed by an investigation of the underlying transition mechanism. Qualitative similarities between wake-induced transition and bypass transition due to free-stream turbulence are discussed and the challenges of predicting boundary-layer transition in this complex environment are pointed out.